Elementary algebra: solving linear equations in one variable  (Page 2/3)

Use the properties of equality to solve: $-5x=-35$ $$\begin{array}{cccc}-5x& =& -35& \\ \frac{-5x}{-5}& =& \frac{-35}{-5}& \mathit{\text{Divide both sides by -5}}\text{.}\hfill \\ x& =& 7& \mathit{\text{Simplify}}\hfill \end{array}$$ The solution set is $\left\{7\right\}$ .

Solve $2x+3=13$ . $$\begin{array}{cccc}2x+3& =& 13& \\ 2x+3-3& =& 13-3& \mathit{\text{Subtract 3 on both sides}}\text{.}\hfill \\ 2x& =& 10& \\ \frac{2x}{2}& =& \frac{10}{2}& \mathit{\text{Divide both sides by 2}}\text{.}\hfill \\ x& =& 5& \end{array}$$ The solution set is $\left\{5\right\}$ .

Solve $-3x-2=9$ . $$\begin{array}{cccc}-3x-2& =& 9& \\ -3x-2+2& =& 9+2& \mathit{\text{Add 2 to both sides}}\text{.}\hfill \\ -3x& =& 11& \\ \frac{-3x}{-3}& =& \frac{11}{-3}& \mathit{\text{Divide both sides by -3}}\text{.}\hfill \\ x& =& -\frac{11}{3}& \end{array}$$ The solution set is $\left\{-\frac{11}{3}\right\}$

Solve $\frac{x}{3}+\frac{1}{2}=\frac{2}{3}$ . $$\begin{array}{cccc}\frac{x}{3}+\frac{1}{2}& =& \frac{2}{3}& \\ \frac{x}{3}+\frac{1}{2}-\frac{1}{2}& =& \frac{2}{3}-\frac{1}{2}& \mathit{\text{Subtract }}\frac{1}{2}\text{ on both sides}\text{.}\hfill \\ \frac{x}{3}& =& \frac{2}{3}\left(\frac{2}{2}\right)-\frac{1}{2}\left(\frac{3}{3}\right)& \\ \frac{x}{3}& =& \frac{4}{6}-\frac{3}{6}& \\ \frac{x}{3}& =& \frac{1}{6}& \\ 3\cdot \frac{x}{3}& =& 3\cdot \frac{1}{6}& \mathit{\text{Multiply both sides by 3}}\text{.}\hfill \\ x& =& \frac{1}{2}& \end{array}$$ The solution set is $\left\{\frac{1}{2}\right\}$ .

Solve $-\frac{4}{5}x-5=15$ . $$\begin{array}{cccc}-\frac{4}{5}x-5& =& 15& \\ -\frac{4}{5}x-5+5& =& 15+5& \mathit{\text{Add 5 on both sides}}\text{.}\hfill \\ -\frac{4}{5}x& =& 20& \\ -\frac{5}{4}\cdot \left(-\frac{4}{5}x\right)& =& -\frac{5}{4}\cdot \left(20\right)& \mathit{\text{Multiply both sides by }}-\frac{5}{4}\text{. }\hfill \\ 1x& =& -5\cdot 5& \mathit{\text{Simplify}}\text{.}\hfill \\ x& =& -25& \end{array}$$ The solution set is $\left\{-25\right\}$ .

Solve for y: $-2y+3=5y+17$ $$\begin{array}{cccc}-2y+3& =& 5y+17& \\ -2y+3-5y& =& 5y+17-5y& \mathit{\text{Subtract 5y on both sides}}\text{.}\hfill \\ -7y+3& =& 17& \\ -7y+3-3& =& 17-3& \mathit{\text{Subtract 3 on both sides}}\text{.}\hfill \\ -7y& =& 14& \\ \frac{-7y}{-7}& =& \frac{14}{-7}& \mathit{\text{Divide both sides by -7}}\text{.}\hfill \\ y& =& -2& \\ & & & \end{array}$$ The solution set is $\left\{-2\right\}$ .